FOR HAMMERSTEIN OPERATORS
IRINA A. LECA
An extension of the Leray-Schauder topological degree is constructed by replacing compact perturbation by composition of mappings of monotone type.
AMS 2000 Subject Classification: 47H11, 47H30.
Key words: degree theory, Hammerstein equations, Nemitsky operators.
Topological degree is the primary tool in the investigation of nonlinear equations. It was introduced by L.E.J. Brouwer in 1912 for mappings on finite- dimensional spaces and generalized by J. Leray and J. Schauder in 1934 to compact perturbations of the identity on Banach infinite-dimensional spaces.
Among the ulterior developments we mention those related to the mappings of type (S). The approaches due to I.V. Skrypnik (1973) and F.E. Browder (1983), as well as those in the monograph of D. O’Regan et al. (2006), are based on the Brouwer degree and use the Galerkin approximation techniques.
A different construction of J. Berkovits and V. Mustonen (1986) combines an elliptic super-regularization method with the Leray-Schauder degree.
We extend the topological degree on real separable Banach spaces to Hammerstein mappings of the form T = I +KN, where K and N are op- erators of monotone type. A recent general result in this direction is due to Berkovits [3]. However, in the original Hammerstein integral equations the pattern nonlinearities generate Nemitsky operators. We pointed out in [9] a coerciveness assumption upon these nonlinearities which bring in the corre- sponding Nemitsky operators of type (S). In this paper, we present a trans- parent description and make precise the classic properties of the Hammerstein topological degree assuming that the nonlinearityN is a determined mapping of type (S). We bring together this degree and the Leray-Schauder principle to obtain an existence result for solutions of an operator Hammerstein equation.
MATH. REPORTS10(60),4 (2008), 327–335
1. NEMITSKY OPERATORS
An integral Hammerstein equation has the form u(x) +
Z
Ω
k(x, y)f(y, u(y)) dy=w(x),
where Ω ⊂ Rn is a domain with σ-finite measure. Assume that the kernel k: Ω×Ω7→Rand the nonlinearityf : Ω×R7→R are measurable functions.
We introduce the linear integral operator Kv(x) =
Z
Ω
k(x, y)v(y) dy,
and we associate with f the Nemitsky operator N u=f(·, u(·)). Then the Hammerstein equation can be briefly written in the operator form
(1.1) u+KN u=w or (I+KN)u=w.
A variety of methods [1] like fixed point or spectral techniques, monotonicity or variational procedures turn out to be useful tools in solving the abstract equations (1.1).
More generally, let X be a real reflexive separable Banach space,X∗ its dual space, let hh, xiorhx, hidenote the value of the functionalh∈X∗ at the elementx∈X. By “→” and “*” we denote the strong and weak convergence, respectively.
Definition 1.1. A mapping F :DF ⊆X 7→X∗ is of type (S) or satisfies condition(S), and we writeF ∈(S), if any sequence{xn}inDF withxn* x for which
lim suphF xn, xn−xi ≤0 is in fact strongly convergent.
Further, we are interested in explicit results concerning the Nemitsky operator on Lebesgue spaces. For instance in [11] is proved the following result.
Let p∈[1,∞) and 1p +p10 = 1. Assume that f satifies the Carath´eodory conditions and there are a constant c >0 and a functionb∈Lp0(Ω)such that
|f(x, r)| ≤c|r|p−1+b(x), ∀(x, r)∈Ω×R.
Then the corresponding Nemitsky operator N = Nf is a bounded continuous operator from Lp(Ω)into Lp0(Ω).
Moreover, we have shown [9] that iff(x, ·),x∈X, is strictly increasing and also satisfies the coerciveness condition
f(x, r)r≥d|r|p+g(x), ∀(x, r)∈Ω×R,
with a constantd >0 andg∈L1(Ω), thenN :Lp(Ω)7→Lp0(Ω) is of type (S).
2. AN EXTENSION OF MAPPINGS OF TYPE (S)
An operator F :DF ⊂X 7→ X is said to bedemicontinuous if xn → x implies F xn * F x. For any bounded open subset D of X, we consider the classes of mappings
LS(D) ={F :D⊂X7→X|F =I+C,whereC:D7→Xis compact}
and
S(D) ={F :D⊂X 7→X∗|Fbounded demicontinuous of type (S)}. A mappingF ∈LS(D) is said to beof Leray-Schauder type. To such a mapping there corresponds the degree function with the same name,LS-degree for short.
The Leray-Schauder degree was extended to mappings of the class S(D) by Berkovits and Mustonen [5].
We now consider a support, that is, a continuous mapping N ∈ S(D), and define a generalization of the operators of type (S).
Definition 2.1. A mappingF :DF ⊆X 7→X isof type (S)N or satisfies condition (S)N, and we writeF ∈(S)N, if for any sequences {xn} ⊂DF and {N xn} ⊂X∗, it follows from xn* x inX,N xn* g inX∗, and
lim suphF xn, N xn−gi ≤0 that xn→x inX.
Denote SN(D) =
F :DF⊆D⊂X7→X|F bounded, demicontinuous of type (S)N . Now, we give a reason for the above definition. First, recall that a bounded operator F :X7→X∗ is quasimonotone, and we write F ∈(QM), if
lim suphF xn, xn−xi ≥0, for any sequence {xn} ⊂DF withxn* x.
Theorem 2.1. Let N ∈S(D) be a continuous operator and K :DK ⊂ X∗ 7→X a quasimonotone, continuous, and bounded mapping such thatN(D)
⊆DK. Then T =I +KN ∈SN(D).
Proof. We have to verify condition SN(D) for the map T = I+KN. Let {xn} ⊂ D and {N xn} be sequences with xn * x, N xn * g and lim suphT xn, N xn−gi ≤ 0. Assume that {xn} does not converge strongly tox. Passing to subsequences, this is equivalent to
lim suphN xn, xn−xi=δ >0.
On the other hand, from the weak convergences xn* x,N xn* g, we have limhN xn−g, xni= limhN xn−g, xn−xi= limhN xn, xn−xi,
so that
lim suphKN xn, N xn−gi= lim suphxn−KN xn, N xn−gi −δ≤ −δ <0, which contradicts the quasimonotonicity of K.
By definition, for any support N ∈ S(D) we have the following pro- perties:
(1)LS(D)⊂SN(D);
(2)KN ∈SN(D) for any bounded demicontinuous mapping K :X∗ 7→
X of type (S);
(3)SN+C(D) =SN(D) for any perturbation C:D7→X∗.
Moreover, when X is a Hilbert space, identifying X with X∗, we have LS(X)⊂S(X).
By Trojansky’s theorem, we may suppose without any loss of generality that X and X∗ are locally uniform convex spaces. In this framework, a good candidate for the support operators are the duality maps J :X 7→ X∗ con- necting the above account with the accretive operator theory [7].
Now, we introduce a generalization of quasimonotonicity.
Definition2.2. A bounded demicontinuous operatorF :DF ⊆X7→Xis said to satisfycondition(QM)N if for any sequences{xn} ⊂DT and{N xn} ⊂ X∗, the conditions xn* x and N xn* g imply
lim suphF xn, N xn−gi ≥0.
It is easy to show that any mapping of type SN(D) satisfies condition (QM)N. By the Calvert-Webb [8] criterion, the operators of type (S) have a regularizing role for quasimonotone mappings. A similar result can be shown for operators satisfying condition (QM)N.
Theorem 2.2. A bounded demicontinuous operator F :X 7→X∗ satis- fies condition (QM)N if and only if F+S∈SN(D) for all S ∈SN(D) and >0.
Proof. LetS ∈SN,F ∈(QM)N,xn* x inX,N xn* g inX∗, and lim suphF xn+Sxn, N xn−gi ≤0.
As F ∈(QM)N, we have lim suphF xn, N xn−gi ≥0,so that lim suphSxn, N xn−gi ≤0,
which implies xn→x, hence F+S∈SN.
Conversely, let F : X 7→ X∗ be a demicontinuous operator such that F +S ∈ SN for all S ∈ SN. In particular, F +I ∈ SN for any > 0. If F /∈ (QM)N then lim suphF xn, N xn−gi = q < 0 for a sequence {xn} and
passing to a subsequence we get limhF xn, N xn−gi=q.
On the other hand, since SN ⊂(QM)N, we have
0≤lim suphF xn+xn, N xn−gi=q+ lim suphxn, N xn−gi, ∀ >0.
As xn,N,g are bounded, we have reached a contradiction.
3. A HAMMERSTEIN DEGREE
In the construction of a topological degree Leray-Schauder for mappings of type (S)N, as in the case of operators of type (S) we use an elliptic super- regularization method of Browder-Ton in a recent form due to Brekovits [2].
As above, let X be a separable reflexive Banach space endowed with a norm such that X and X∗ are locally uniformly convex.
Browder-Ton Theorem. There exists a real separable Hilbert space and a compact linear injection Φ :H7→X∗ such thatΦ (H) is dense inX∗.
We define the adjoint operator by
(Φ∗x, u) =hx,Φui, ∀x∈X, u∈H,
where (·, ·) stands for the inner product in H. Since Φ (H) is dense in X∗, the mapping Φ∗:X 7→H also is a linear compact injection.
Let N ∈ S(D) be a fixed operator as above. With any F ∈ SN(D) we associate the family of mappings {Fλ|λ >0} defined by
(3.1) Fλ=N+λΦΦ∗F, ∀λ >0.
For anyλ >0,Fλ mapsDintoXand has the formN+Cλ, whereCλ= ΦΦ∗F is a compact mapping. HenceN+Cλ∈S(D) and theS-degreedS makes sense for the triple (Fλ, D, y) whenevery /∈Fλ(∂D) ([5], [11]).
Proposition 3.1. Let F ∈ SN(D) and assume there are sequences {xn} ⊂ D and {λn} ⊂ R+ such that Fλnxn = 0 as λn → ∞. Then there exists x∈Dand a subsequence {xj} such that xj →x and F x= 0.
Proof. Passing to subsequences, we may assume that xn * x, F xn * w in X and N xn * g in X∗. Since ΦΦ∗ is linear and compact, we have ΦΦ∗F xn→ΦΦ∗w inX∗.
On the other hand, Fλnxn = 0 implies ΦΦ∗F xn = −λ−1n N xn → 0 as n→ ∞. We get ΦΦ∗w= 0 and, therefore,w= 0. Besides,
lim suphF xn, N xn−gi= lim suphF xn, N xni=
= lim suphF xn,−λnΦΦ∗F xni= lim sup −λnkΦ∗F xnk2H
≤0,
where k · kH is the norm in H. As F ∈ SN(D), we have xn → x ∈ D and, by demicontinuity, F xn* F x inX. By the uniqueness of the weak limit, we conclude that F x= 0.
Corollary 3.1. Let F ∈ SN(D) and assume that 0 ∈/ F(A), where A ⊂D is a closed subset. Then there exists ˜λ > 0 such that 0 ∈/ Fλ(A) for λ≥˜λ. Moreover, if 0∈/ F(∂D) there exists λ0 >0 such that dS(Fλ, D,0) is constant for λ≥λ0.
Proof. Otherwise, there are sequences {λn} ⊂ R+ and {xn} ⊂ A such that λn → ∞ and Fλnxn = 0. Proposition 3.1 ensures the existence of a solution x ∈ A of the equation F x = 0, contradicting an assumption made.
Therefore, there exists ˜λ >0 such thatFλx6= 0 for λ≥λ.˜
For the second assertion, we assume that λ0 = ˜λ for A = ∂D. Hence, dS(Fλ, D,0) is well-defined for all λ≥λ0. Forλ1 > λ2≥λ0, we consider the S-homotopy
tFλ1+ (1−t)Fλ2 =N + (tλ1+ (1−t)λ2) ΦΦ∗F =Fλt,
where λt = tλ1 + (1−t)λ2 and 0 ≤ t ≤ 1. Since λ1 ≥ λt ≥ λ2 ≥ λ0, we have Fλtx 6= 0 for all x ∈ ∂D and t ∈ [0,1]. By the homotopy property of the S-degree, we have dS(Fλ1, D,0) =dS(Fλ2, D,0),thas completing the proof.
On account of Corollary 3.1, it is relevant to consider below this common value of dS(Fλ, D,0) forλ >0 sufficiently large.
Definition 3.1. For a triple (F, D,0) with F ∈SN(D) the Hammerstein degree DN is defined as
DN(F, D,0) =dS(Fλ, D,0), λ≥λ0, whenever 0∈/ F(∂D). Moreover, for any y∈X\F(∂D),
DN(F, D, y) =DN(F−y, D,0).
This degree mapping refers to an arbitrarily chosen elementN ∈S(D).
4. PROPERTIES OF THE DEGREE
Now, we show that the integer-valued function DN introduced above in the class of mappingsSN(D), verifies the properties of the classical topological degree. Similarly to Definition 2.1, we say that a homotopyH : [0,1]×DH 7→
X is oftype (S)N or satisfiescondition(S)N if for any sequences{tn} ⊂[0,1], {xn} ⊂DH and {N xn} ⊂X∗, it follows fromtn→t,xn* xinX,N xn* g in X∗ and
lim suphH(tn, xn), N xn−gi ≤0
that xn→x and H(tn, xn)* H(t, x) inX. Moreover, similarly to represen- tation (3.1), we also have
Hλ(t, ·) =N+λΦΦ∗H(t, ·), ∀λ >0,
where Hλ(t, ·) defines a homotopy oftype (S) for any fixedλ >0. It is easy to establish for H(t, ·) an assertion similar to Corollary 3.1.
As expected, the new degree mapping follows immediately from the prop- erties of the S-degree.
(1)Normalization. Condition (S) on N implies that the identity map I belongs to SN(D). Moreover,DN(I, D, h) = 1 for allh∈D.
(2) Domain additivity. If D1, D2 are disjoint open subsets of D and h /∈F D\(D1∪D2)
, thenDN(F, D, h) =DN(F, D1, h) +DN(F, D2, h).
(3) Homotopy invariance. Let F ∈ SN(D) and h : [0,1] → X be a continuous curve such that ht ∈/ H(t, ∂D) for all t in [0,1]. Then the value DN(H(t, ·), D, ht) is constant fort∈[0,1].
(4)Existence of solution. By a substraction, we can takeh= 0. It follows by Corolarry 3.1 that 0∈/Fλ(D) for allλ≥λ. But˜ dS(Fλ, D,0) = 0 forλ≥˜λ implies DN(F, D,0) = 0. Therefore,DN(F, D,0)6= 0 yields 0∈F(D).
Other properties of DN are inherited from those of dS. Furthermore, the uniqueness of extensions of degree mappings have been recently discussed in [4].
5. AN EXISTENCE RESULT
The existence of solutions of operator Hammerstein equations relies on the Leray-Schauder Principle. Let X be a Banach space, D a bounded domain in X with 0∈D, and C :X → X a compact operator. Assume that u+tCu6= 0 for all u∈∂D,0< t <1.ThenI+C has at least one zero inD.
A more general form of this result can be found in ([13], Ch. 4). The homotopy invariance of the new degree DN allows the extension below.
Corollary 6.1. Assume F ∈SN D
and h∈D. If 0∈/F(∂D) and (6.1) (1−t)u+tF u6= (1−t)h
for all x∈∂D, 0< t <1, then the equation F u= 0 has at least one solution in D.
Hypothesis (6.1) is the nonvanishing condition for the affine homotopy linking the identity andF. In particular, when 0∈Dcondition (6.1) becomes
(6.2) t(u−F u)6=u
whenever u ∈ ∂D, 0 < t < 1. Replacing F by the Hammerstein mapping T =I+KN, we get
(6.3) u+tKN u6= 0
whenever u∈∂D, 0< t <1.
Theorem 6.1. Let K, N be as in Theorem2.1. Suppose that K andN verify the coerciveness conditions
(Kw, w)
kwk → ∞ as kwk → ∞, w∈X∗, and kN uk → ∞ as kuk → ∞.
In addition, assume that there is r >0 such that hN u, ui ≥0 for all kuk ≥r.
Then the Hammerstein equation T u=f has solutions for anyf ∈X.
Proof. By a translation, we may assume thatf = 0.
On the other hand, the coerciveness of K implies that there is r1 > 0 such that hKw,wikwk ≥1, whenkwk ≥r1and sohKw, wi ≥r1 >0 whenkwk ≥r1. Now, using the hypothesis on N, there is r2 > r such that kN uk > r1 for kuk ≥r2. We deduce thathKN u, N ui>0 if kuk ≥r2. Consequently,
hN u, T ui=hN u, ui+thKN u, N ui>0
for all kuk ≥r2, 0< t <1. By of the Leray-Schauder principle (Corollary 6.1 and condition (6.3)), the equation T u = f has solutions in the ball D with the centre at the origin with radius larger than r2.
Acknowledgements. The author wants to express her gratitude to Professor Dan D. Pascali for his support in the preparation of this paper and to Dr. Cristian Bereanu (Univ. Catholique de Louvain) for references on existence results.
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Received 2 June 2008 Naval Academy “Mircea cel B˘atrˆan”
Fulger Street no. 1 900394 Constant¸a, Romania
irinaleca@hotmail.com
